3.5.16 \(\int x (a+b \log (c (d+e \sqrt {x})^n))^3 \, dx\) [416]

Optimal. Leaf size=595 \[ -\frac {9 b^3 d^2 n^3 \left (d+e \sqrt {x}\right )^2}{4 e^4}+\frac {4 b^3 d n^3 \left (d+e \sqrt {x}\right )^3}{9 e^4}-\frac {3 b^3 n^3 \left (d+e \sqrt {x}\right )^4}{64 e^4}-\frac {12 a b^2 d^3 n^2 \sqrt {x}}{e^3}+\frac {12 b^3 d^3 n^3 \sqrt {x}}{e^3}-\frac {12 b^3 d^3 n^2 \left (d+e \sqrt {x}\right ) \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{e^4}+\frac {9 b^2 d^2 n^2 \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{2 e^4}-\frac {4 b^2 d n^2 \left (d+e \sqrt {x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{3 e^4}+\frac {3 b^2 n^2 \left (d+e \sqrt {x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{16 e^4}+\frac {6 b d^3 n \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{e^4}-\frac {9 b d^2 n \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{2 e^4}+\frac {2 b d n \left (d+e \sqrt {x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{e^4}-\frac {3 b n \left (d+e \sqrt {x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{8 e^4}-\frac {2 d^3 \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^4}+\frac {3 d^2 \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^4}-\frac {2 d \left (d+e \sqrt {x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^4}+\frac {\left (d+e \sqrt {x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{2 e^4} \]

[Out]

-12*a*b^2*d^3*n^2*x^(1/2)/e^3+12*b^3*d^3*n^3*x^(1/2)/e^3-12*b^3*d^3*n^2*ln(c*(d+e*x^(1/2))^n)*(d+e*x^(1/2))/e^
4+6*b*d^3*n*(a+b*ln(c*(d+e*x^(1/2))^n))^2*(d+e*x^(1/2))/e^4-2*d^3*(a+b*ln(c*(d+e*x^(1/2))^n))^3*(d+e*x^(1/2))/
e^4-9/4*b^3*d^2*n^3*(d+e*x^(1/2))^2/e^4+9/2*b^2*d^2*n^2*(a+b*ln(c*(d+e*x^(1/2))^n))*(d+e*x^(1/2))^2/e^4-9/2*b*
d^2*n*(a+b*ln(c*(d+e*x^(1/2))^n))^2*(d+e*x^(1/2))^2/e^4+3*d^2*(a+b*ln(c*(d+e*x^(1/2))^n))^3*(d+e*x^(1/2))^2/e^
4+4/9*b^3*d*n^3*(d+e*x^(1/2))^3/e^4-4/3*b^2*d*n^2*(a+b*ln(c*(d+e*x^(1/2))^n))*(d+e*x^(1/2))^3/e^4+2*b*d*n*(a+b
*ln(c*(d+e*x^(1/2))^n))^2*(d+e*x^(1/2))^3/e^4-2*d*(a+b*ln(c*(d+e*x^(1/2))^n))^3*(d+e*x^(1/2))^3/e^4-3/64*b^3*n
^3*(d+e*x^(1/2))^4/e^4+3/16*b^2*n^2*(a+b*ln(c*(d+e*x^(1/2))^n))*(d+e*x^(1/2))^4/e^4-3/8*b*n*(a+b*ln(c*(d+e*x^(
1/2))^n))^2*(d+e*x^(1/2))^4/e^4+1/2*(a+b*ln(c*(d+e*x^(1/2))^n))^3*(d+e*x^(1/2))^4/e^4

________________________________________________________________________________________

Rubi [A]
time = 0.40, antiderivative size = 595, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {2504, 2448, 2436, 2333, 2332, 2437, 2342, 2341} \begin {gather*} \frac {9 b^2 d^2 n^2 \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{2 e^4}+\frac {3 b^2 n^2 \left (d+e \sqrt {x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{16 e^4}-\frac {4 b^2 d n^2 \left (d+e \sqrt {x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{3 e^4}-\frac {12 a b^2 d^3 n^2 \sqrt {x}}{e^3}-\frac {2 d^3 \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^4}+\frac {6 b d^3 n \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{e^4}+\frac {3 d^2 \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^4}-\frac {9 b d^2 n \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{2 e^4}+\frac {\left (d+e \sqrt {x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{2 e^4}-\frac {3 b n \left (d+e \sqrt {x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{8 e^4}-\frac {2 d \left (d+e \sqrt {x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^4}+\frac {2 b d n \left (d+e \sqrt {x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{e^4}-\frac {12 b^3 d^3 n^2 \left (d+e \sqrt {x}\right ) \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{e^4}+\frac {12 b^3 d^3 n^3 \sqrt {x}}{e^3}-\frac {9 b^3 d^2 n^3 \left (d+e \sqrt {x}\right )^2}{4 e^4}-\frac {3 b^3 n^3 \left (d+e \sqrt {x}\right )^4}{64 e^4}+\frac {4 b^3 d n^3 \left (d+e \sqrt {x}\right )^3}{9 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*(a + b*Log[c*(d + e*Sqrt[x])^n])^3,x]

[Out]

(-9*b^3*d^2*n^3*(d + e*Sqrt[x])^2)/(4*e^4) + (4*b^3*d*n^3*(d + e*Sqrt[x])^3)/(9*e^4) - (3*b^3*n^3*(d + e*Sqrt[
x])^4)/(64*e^4) - (12*a*b^2*d^3*n^2*Sqrt[x])/e^3 + (12*b^3*d^3*n^3*Sqrt[x])/e^3 - (12*b^3*d^3*n^2*(d + e*Sqrt[
x])*Log[c*(d + e*Sqrt[x])^n])/e^4 + (9*b^2*d^2*n^2*(d + e*Sqrt[x])^2*(a + b*Log[c*(d + e*Sqrt[x])^n]))/(2*e^4)
- (4*b^2*d*n^2*(d + e*Sqrt[x])^3*(a + b*Log[c*(d + e*Sqrt[x])^n]))/(3*e^4) + (3*b^2*n^2*(d + e*Sqrt[x])^4*(a
+ b*Log[c*(d + e*Sqrt[x])^n]))/(16*e^4) + (6*b*d^3*n*(d + e*Sqrt[x])*(a + b*Log[c*(d + e*Sqrt[x])^n])^2)/e^4 -
(9*b*d^2*n*(d + e*Sqrt[x])^2*(a + b*Log[c*(d + e*Sqrt[x])^n])^2)/(2*e^4) + (2*b*d*n*(d + e*Sqrt[x])^3*(a + b*
Log[c*(d + e*Sqrt[x])^n])^2)/e^4 - (3*b*n*(d + e*Sqrt[x])^4*(a + b*Log[c*(d + e*Sqrt[x])^n])^2)/(8*e^4) - (2*d
^3*(d + e*Sqrt[x])*(a + b*Log[c*(d + e*Sqrt[x])^n])^3)/e^4 + (3*d^2*(d + e*Sqrt[x])^2*(a + b*Log[c*(d + e*Sqrt
[x])^n])^3)/e^4 - (2*d*(d + e*Sqrt[x])^3*(a + b*Log[c*(d + e*Sqrt[x])^n])^3)/e^4 + ((d + e*Sqrt[x])^4*(a + b*L
og[c*(d + e*Sqrt[x])^n])^3)/(2*e^4)

Rule 2332

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2333

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]

Rule 2341

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^
n])/(d*(m + 1))), x] - Simp[b*n*((d*x)^(m + 1)/(d*(m + 1)^2)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2342

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Lo
g[c*x^n])^p/(d*(m + 1))), x] - Dist[b*n*(p/(m + 1)), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]

Rule 2436

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]

Rule 2437

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[(f*(x/d))^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
&& EqQ[e*f - d*g, 0]

Rule 2448

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Int[Exp
andIntegrand[(f + g*x)^q*(a + b*Log[c*(d + e*x)^n])^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && NeQ[
e*f - d*g, 0] && IGtQ[q, 0]

Rule 2504

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p,
q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) &&  !(EqQ[q, 1] && ILtQ[n, 0] &&
IGtQ[m, 0])

Rubi steps

\begin {align*} \int x \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3 \, dx &=2 \text {Subst}\left (\int x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\sqrt {x}\right )\\ &=2 \text {Subst}\left (\int \left (-\frac {d^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^3}+\frac {3 d^2 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^3}-\frac {3 d (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^3}+\frac {(d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^3}\right ) \, dx,x,\sqrt {x}\right )\\ &=\frac {2 \text {Subst}\left (\int (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\sqrt {x}\right )}{e^3}-\frac {(6 d) \text {Subst}\left (\int (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\sqrt {x}\right )}{e^3}+\frac {\left (6 d^2\right ) \text {Subst}\left (\int (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\sqrt {x}\right )}{e^3}-\frac {\left (2 d^3\right ) \text {Subst}\left (\int \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\sqrt {x}\right )}{e^3}\\ &=\frac {2 \text {Subst}\left (\int x^3 \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e \sqrt {x}\right )}{e^4}-\frac {(6 d) \text {Subst}\left (\int x^2 \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e \sqrt {x}\right )}{e^4}+\frac {\left (6 d^2\right ) \text {Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e \sqrt {x}\right )}{e^4}-\frac {\left (2 d^3\right ) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e \sqrt {x}\right )}{e^4}\\ &=-\frac {2 d^3 \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^4}+\frac {3 d^2 \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^4}-\frac {2 d \left (d+e \sqrt {x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^4}+\frac {\left (d+e \sqrt {x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{2 e^4}-\frac {(3 b n) \text {Subst}\left (\int x^3 \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e \sqrt {x}\right )}{2 e^4}+\frac {(6 b d n) \text {Subst}\left (\int x^2 \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e \sqrt {x}\right )}{e^4}-\frac {\left (9 b d^2 n\right ) \text {Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e \sqrt {x}\right )}{e^4}+\frac {\left (6 b d^3 n\right ) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e \sqrt {x}\right )}{e^4}\\ &=\frac {6 b d^3 n \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{e^4}-\frac {9 b d^2 n \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{2 e^4}+\frac {2 b d n \left (d+e \sqrt {x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{e^4}-\frac {3 b n \left (d+e \sqrt {x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{8 e^4}-\frac {2 d^3 \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^4}+\frac {3 d^2 \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^4}-\frac {2 d \left (d+e \sqrt {x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^4}+\frac {\left (d+e \sqrt {x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{2 e^4}+\frac {\left (3 b^2 n^2\right ) \text {Subst}\left (\int x^3 \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e \sqrt {x}\right )}{4 e^4}-\frac {\left (4 b^2 d n^2\right ) \text {Subst}\left (\int x^2 \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e \sqrt {x}\right )}{e^4}+\frac {\left (9 b^2 d^2 n^2\right ) \text {Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e \sqrt {x}\right )}{e^4}-\frac {\left (12 b^2 d^3 n^2\right ) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e \sqrt {x}\right )}{e^4}\\ &=-\frac {9 b^3 d^2 n^3 \left (d+e \sqrt {x}\right )^2}{4 e^4}+\frac {4 b^3 d n^3 \left (d+e \sqrt {x}\right )^3}{9 e^4}-\frac {3 b^3 n^3 \left (d+e \sqrt {x}\right )^4}{64 e^4}-\frac {12 a b^2 d^3 n^2 \sqrt {x}}{e^3}+\frac {9 b^2 d^2 n^2 \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{2 e^4}-\frac {4 b^2 d n^2 \left (d+e \sqrt {x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{3 e^4}+\frac {3 b^2 n^2 \left (d+e \sqrt {x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{16 e^4}+\frac {6 b d^3 n \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{e^4}-\frac {9 b d^2 n \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{2 e^4}+\frac {2 b d n \left (d+e \sqrt {x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{e^4}-\frac {3 b n \left (d+e \sqrt {x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{8 e^4}-\frac {2 d^3 \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^4}+\frac {3 d^2 \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^4}-\frac {2 d \left (d+e \sqrt {x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^4}+\frac {\left (d+e \sqrt {x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{2 e^4}-\frac {\left (12 b^3 d^3 n^2\right ) \text {Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e \sqrt {x}\right )}{e^4}\\ &=-\frac {9 b^3 d^2 n^3 \left (d+e \sqrt {x}\right )^2}{4 e^4}+\frac {4 b^3 d n^3 \left (d+e \sqrt {x}\right )^3}{9 e^4}-\frac {3 b^3 n^3 \left (d+e \sqrt {x}\right )^4}{64 e^4}-\frac {12 a b^2 d^3 n^2 \sqrt {x}}{e^3}+\frac {12 b^3 d^3 n^3 \sqrt {x}}{e^3}-\frac {12 b^3 d^3 n^2 \left (d+e \sqrt {x}\right ) \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{e^4}+\frac {9 b^2 d^2 n^2 \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{2 e^4}-\frac {4 b^2 d n^2 \left (d+e \sqrt {x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{3 e^4}+\frac {3 b^2 n^2 \left (d+e \sqrt {x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{16 e^4}+\frac {6 b d^3 n \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{e^4}-\frac {9 b d^2 n \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{2 e^4}+\frac {2 b d n \left (d+e \sqrt {x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{e^4}-\frac {3 b n \left (d+e \sqrt {x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{8 e^4}-\frac {2 d^3 \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^4}+\frac {3 d^2 \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^4}-\frac {2 d \left (d+e \sqrt {x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^4}+\frac {\left (d+e \sqrt {x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{2 e^4}\\ \end {align*}

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Mathematica [A]
time = 0.32, size = 517, normalized size = 0.87 \begin {gather*} \frac {-288 b^3 d^4 n^3 \log ^3\left (d+e \sqrt {x}\right )+72 b^2 d^4 n^2 \log ^2\left (d+e \sqrt {x}\right ) \left (12 a-25 b n+12 b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )-12 b d^4 n \log \left (d+e \sqrt {x}\right ) \left (72 a^2-300 a b n+415 b^2 n^2+12 b (12 a-25 b n) \log \left (c \left (d+e \sqrt {x}\right )^n\right )+72 b^2 \log ^2\left (c \left (d+e \sqrt {x}\right )^n\right )\right )+e \sqrt {x} \left (288 a^3 e^3 x^{3/2}+b^3 n^3 \left (4980 d^3-690 d^2 e \sqrt {x}+148 d e^2 x-27 e^3 x^{3/2}\right )-12 a b^2 n^2 \left (300 d^3-78 d^2 e \sqrt {x}+28 d e^2 x-9 e^3 x^{3/2}\right )+72 a^2 b n \left (12 d^3-6 d^2 e \sqrt {x}+4 d e^2 x-3 e^3 x^{3/2}\right )+12 b \left (72 a^2 e^3 x^{3/2}+12 a b n \left (12 d^3-6 d^2 e \sqrt {x}+4 d e^2 x-3 e^3 x^{3/2}\right )+b^2 n^2 \left (-300 d^3+78 d^2 e \sqrt {x}-28 d e^2 x+9 e^3 x^{3/2}\right )\right ) \log \left (c \left (d+e \sqrt {x}\right )^n\right )+72 b^2 \left (12 a e^3 x^{3/2}+b n \left (12 d^3-6 d^2 e \sqrt {x}+4 d e^2 x-3 e^3 x^{3/2}\right )\right ) \log ^2\left (c \left (d+e \sqrt {x}\right )^n\right )+288 b^3 e^3 x^{3/2} \log ^3\left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{576 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*Log[c*(d + e*Sqrt[x])^n])^3,x]

[Out]

(-288*b^3*d^4*n^3*Log[d + e*Sqrt[x]]^3 + 72*b^2*d^4*n^2*Log[d + e*Sqrt[x]]^2*(12*a - 25*b*n + 12*b*Log[c*(d +
e*Sqrt[x])^n]) - 12*b*d^4*n*Log[d + e*Sqrt[x]]*(72*a^2 - 300*a*b*n + 415*b^2*n^2 + 12*b*(12*a - 25*b*n)*Log[c*
(d + e*Sqrt[x])^n] + 72*b^2*Log[c*(d + e*Sqrt[x])^n]^2) + e*Sqrt[x]*(288*a^3*e^3*x^(3/2) + b^3*n^3*(4980*d^3 -
690*d^2*e*Sqrt[x] + 148*d*e^2*x - 27*e^3*x^(3/2)) - 12*a*b^2*n^2*(300*d^3 - 78*d^2*e*Sqrt[x] + 28*d*e^2*x - 9
*e^3*x^(3/2)) + 72*a^2*b*n*(12*d^3 - 6*d^2*e*Sqrt[x] + 4*d*e^2*x - 3*e^3*x^(3/2)) + 12*b*(72*a^2*e^3*x^(3/2) +
12*a*b*n*(12*d^3 - 6*d^2*e*Sqrt[x] + 4*d*e^2*x - 3*e^3*x^(3/2)) + b^2*n^2*(-300*d^3 + 78*d^2*e*Sqrt[x] - 28*d
*e^2*x + 9*e^3*x^(3/2)))*Log[c*(d + e*Sqrt[x])^n] + 72*b^2*(12*a*e^3*x^(3/2) + b*n*(12*d^3 - 6*d^2*e*Sqrt[x] +
4*d*e^2*x - 3*e^3*x^(3/2)))*Log[c*(d + e*Sqrt[x])^n]^2 + 288*b^3*e^3*x^(3/2)*Log[c*(d + e*Sqrt[x])^n]^3))/(57
6*e^4)

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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int x \left (a +b \ln \left (c \left (d +e \sqrt {x}\right )^{n}\right )\right )^{3}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a+b*ln(c*(d+e*x^(1/2))^n))^3,x)

[Out]

int(x*(a+b*ln(c*(d+e*x^(1/2))^n))^3,x)

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Maxima [A]
time = 0.31, size = 537, normalized size = 0.90 \begin {gather*} \frac {1}{2} \, b^{3} x^{2} \log \left ({\left (\sqrt {x} e + d\right )}^{n} c\right )^{3} + \frac {3}{2} \, a b^{2} x^{2} \log \left ({\left (\sqrt {x} e + d\right )}^{n} c\right )^{2} - \frac {1}{8} \, {\left (12 \, d^{4} e^{\left (-5\right )} \log \left (\sqrt {x} e + d\right ) + {\left (6 \, d^{2} x e - 12 \, d^{3} \sqrt {x} - 4 \, d x^{\frac {3}{2}} e^{2} + 3 \, x^{2} e^{3}\right )} e^{\left (-4\right )}\right )} a^{2} b n e + \frac {3}{2} \, a^{2} b x^{2} \log \left ({\left (\sqrt {x} e + d\right )}^{n} c\right ) + \frac {1}{2} \, a^{3} x^{2} + \frac {1}{48} \, {\left ({\left (72 \, d^{4} \log \left (\sqrt {x} e + d\right )^{2} + 300 \, d^{4} \log \left (\sqrt {x} e + d\right ) - 300 \, d^{3} \sqrt {x} e + 78 \, d^{2} x e^{2} - 28 \, d x^{\frac {3}{2}} e^{3} + 9 \, x^{2} e^{4}\right )} n^{2} e^{\left (-4\right )} - 12 \, {\left (12 \, d^{4} e^{\left (-5\right )} \log \left (\sqrt {x} e + d\right ) + {\left (6 \, d^{2} x e - 12 \, d^{3} \sqrt {x} - 4 \, d x^{\frac {3}{2}} e^{2} + 3 \, x^{2} e^{3}\right )} e^{\left (-4\right )}\right )} n e \log \left ({\left (\sqrt {x} e + d\right )}^{n} c\right )\right )} a b^{2} - \frac {1}{576} \, {\left (72 \, {\left (12 \, d^{4} e^{\left (-5\right )} \log \left (\sqrt {x} e + d\right ) + {\left (6 \, d^{2} x e - 12 \, d^{3} \sqrt {x} - 4 \, d x^{\frac {3}{2}} e^{2} + 3 \, x^{2} e^{3}\right )} e^{\left (-4\right )}\right )} n e \log \left ({\left (\sqrt {x} e + d\right )}^{n} c\right )^{2} + {\left ({\left (288 \, d^{4} \log \left (\sqrt {x} e + d\right )^{3} + 1800 \, d^{4} \log \left (\sqrt {x} e + d\right )^{2} + 4980 \, d^{4} \log \left (\sqrt {x} e + d\right ) - 4980 \, d^{3} \sqrt {x} e + 690 \, d^{2} x e^{2} - 148 \, d x^{\frac {3}{2}} e^{3} + 27 \, x^{2} e^{4}\right )} n^{2} e^{\left (-5\right )} - 12 \, {\left (72 \, d^{4} \log \left (\sqrt {x} e + d\right )^{2} + 300 \, d^{4} \log \left (\sqrt {x} e + d\right ) - 300 \, d^{3} \sqrt {x} e + 78 \, d^{2} x e^{2} - 28 \, d x^{\frac {3}{2}} e^{3} + 9 \, x^{2} e^{4}\right )} n e^{\left (-5\right )} \log \left ({\left (\sqrt {x} e + d\right )}^{n} c\right )\right )} n e\right )} b^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*log(c*(d+e*x^(1/2))^n))^3,x, algorithm="maxima")

[Out]

1/2*b^3*x^2*log((sqrt(x)*e + d)^n*c)^3 + 3/2*a*b^2*x^2*log((sqrt(x)*e + d)^n*c)^2 - 1/8*(12*d^4*e^(-5)*log(sqr
t(x)*e + d) + (6*d^2*x*e - 12*d^3*sqrt(x) - 4*d*x^(3/2)*e^2 + 3*x^2*e^3)*e^(-4))*a^2*b*n*e + 3/2*a^2*b*x^2*log
((sqrt(x)*e + d)^n*c) + 1/2*a^3*x^2 + 1/48*((72*d^4*log(sqrt(x)*e + d)^2 + 300*d^4*log(sqrt(x)*e + d) - 300*d^
3*sqrt(x)*e + 78*d^2*x*e^2 - 28*d*x^(3/2)*e^3 + 9*x^2*e^4)*n^2*e^(-4) - 12*(12*d^4*e^(-5)*log(sqrt(x)*e + d) +
(6*d^2*x*e - 12*d^3*sqrt(x) - 4*d*x^(3/2)*e^2 + 3*x^2*e^3)*e^(-4))*n*e*log((sqrt(x)*e + d)^n*c))*a*b^2 - 1/57
6*(72*(12*d^4*e^(-5)*log(sqrt(x)*e + d) + (6*d^2*x*e - 12*d^3*sqrt(x) - 4*d*x^(3/2)*e^2 + 3*x^2*e^3)*e^(-4))*n
*e*log((sqrt(x)*e + d)^n*c)^2 + ((288*d^4*log(sqrt(x)*e + d)^3 + 1800*d^4*log(sqrt(x)*e + d)^2 + 4980*d^4*log(
sqrt(x)*e + d) - 4980*d^3*sqrt(x)*e + 690*d^2*x*e^2 - 148*d*x^(3/2)*e^3 + 27*x^2*e^4)*n^2*e^(-5) - 12*(72*d^4*
log(sqrt(x)*e + d)^2 + 300*d^4*log(sqrt(x)*e + d) - 300*d^3*sqrt(x)*e + 78*d^2*x*e^2 - 28*d*x^(3/2)*e^3 + 9*x^
2*e^4)*n*e^(-5)*log((sqrt(x)*e + d)^n*c))*n*e)*b^3

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Fricas [A]
time = 0.45, size = 798, normalized size = 1.34 \begin {gather*} \frac {1}{576} \, {\left (288 \, b^{3} x^{2} e^{4} \log \left (c\right )^{3} - 9 \, {\left (3 \, b^{3} n^{3} - 12 \, a b^{2} n^{2} + 24 \, a^{2} b n - 32 \, a^{3}\right )} x^{2} e^{4} - 288 \, {\left (b^{3} d^{4} n^{3} - b^{3} n^{3} x^{2} e^{4}\right )} \log \left (\sqrt {x} e + d\right )^{3} - 6 \, {\left (115 \, b^{3} d^{2} n^{3} - 156 \, a b^{2} d^{2} n^{2} + 72 \, a^{2} b d^{2} n\right )} x e^{2} + 72 \, {\left (25 \, b^{3} d^{4} n^{3} - 6 \, b^{3} d^{2} n^{3} x e^{2} - 12 \, a b^{2} d^{4} n^{2} - 3 \, {\left (b^{3} n^{3} - 4 \, a b^{2} n^{2}\right )} x^{2} e^{4} - 12 \, {\left (b^{3} d^{4} n^{2} - b^{3} n^{2} x^{2} e^{4}\right )} \log \left (c\right ) + 4 \, {\left (3 \, b^{3} d^{3} n^{3} e + b^{3} d n^{3} x e^{3}\right )} \sqrt {x}\right )} \log \left (\sqrt {x} e + d\right )^{2} - 216 \, {\left (2 \, b^{3} d^{2} n x e^{2} + {\left (b^{3} n - 4 \, a b^{2}\right )} x^{2} e^{4}\right )} \log \left (c\right )^{2} - 12 \, {\left (415 \, b^{3} d^{4} n^{3} - 300 \, a b^{2} d^{4} n^{2} + 72 \, a^{2} b d^{4} n - 9 \, {\left (b^{3} n^{3} - 4 \, a b^{2} n^{2} + 8 \, a^{2} b n\right )} x^{2} e^{4} - 6 \, {\left (13 \, b^{3} d^{2} n^{3} - 12 \, a b^{2} d^{2} n^{2}\right )} x e^{2} + 72 \, {\left (b^{3} d^{4} n - b^{3} n x^{2} e^{4}\right )} \log \left (c\right )^{2} - 12 \, {\left (25 \, b^{3} d^{4} n^{2} - 6 \, b^{3} d^{2} n^{2} x e^{2} - 12 \, a b^{2} d^{4} n - 3 \, {\left (b^{3} n^{2} - 4 \, a b^{2} n\right )} x^{2} e^{4}\right )} \log \left (c\right ) + 4 \, {\left ({\left (7 \, b^{3} d n^{3} - 12 \, a b^{2} d n^{2}\right )} x e^{3} + 3 \, {\left (25 \, b^{3} d^{3} n^{3} - 12 \, a b^{2} d^{3} n^{2}\right )} e - 12 \, {\left (3 \, b^{3} d^{3} n^{2} e + b^{3} d n^{2} x e^{3}\right )} \log \left (c\right )\right )} \sqrt {x}\right )} \log \left (\sqrt {x} e + d\right ) + 36 \, {\left (3 \, {\left (b^{3} n^{2} - 4 \, a b^{2} n + 8 \, a^{2} b\right )} x^{2} e^{4} + 2 \, {\left (13 \, b^{3} d^{2} n^{2} - 12 \, a b^{2} d^{2} n\right )} x e^{2}\right )} \log \left (c\right ) + 4 \, {\left ({\left (37 \, b^{3} d n^{3} - 84 \, a b^{2} d n^{2} + 72 \, a^{2} b d n\right )} x e^{3} + 72 \, {\left (3 \, b^{3} d^{3} n e + b^{3} d n x e^{3}\right )} \log \left (c\right )^{2} + 3 \, {\left (415 \, b^{3} d^{3} n^{3} - 300 \, a b^{2} d^{3} n^{2} + 72 \, a^{2} b d^{3} n\right )} e - 12 \, {\left ({\left (7 \, b^{3} d n^{2} - 12 \, a b^{2} d n\right )} x e^{3} + 3 \, {\left (25 \, b^{3} d^{3} n^{2} - 12 \, a b^{2} d^{3} n\right )} e\right )} \log \left (c\right )\right )} \sqrt {x}\right )} e^{\left (-4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*log(c*(d+e*x^(1/2))^n))^3,x, algorithm="fricas")

[Out]

1/576*(288*b^3*x^2*e^4*log(c)^3 - 9*(3*b^3*n^3 - 12*a*b^2*n^2 + 24*a^2*b*n - 32*a^3)*x^2*e^4 - 288*(b^3*d^4*n^
3 - b^3*n^3*x^2*e^4)*log(sqrt(x)*e + d)^3 - 6*(115*b^3*d^2*n^3 - 156*a*b^2*d^2*n^2 + 72*a^2*b*d^2*n)*x*e^2 + 7
2*(25*b^3*d^4*n^3 - 6*b^3*d^2*n^3*x*e^2 - 12*a*b^2*d^4*n^2 - 3*(b^3*n^3 - 4*a*b^2*n^2)*x^2*e^4 - 12*(b^3*d^4*n
^2 - b^3*n^2*x^2*e^4)*log(c) + 4*(3*b^3*d^3*n^3*e + b^3*d*n^3*x*e^3)*sqrt(x))*log(sqrt(x)*e + d)^2 - 216*(2*b^
3*d^2*n*x*e^2 + (b^3*n - 4*a*b^2)*x^2*e^4)*log(c)^2 - 12*(415*b^3*d^4*n^3 - 300*a*b^2*d^4*n^2 + 72*a^2*b*d^4*n
- 9*(b^3*n^3 - 4*a*b^2*n^2 + 8*a^2*b*n)*x^2*e^4 - 6*(13*b^3*d^2*n^3 - 12*a*b^2*d^2*n^2)*x*e^2 + 72*(b^3*d^4*n
- b^3*n*x^2*e^4)*log(c)^2 - 12*(25*b^3*d^4*n^2 - 6*b^3*d^2*n^2*x*e^2 - 12*a*b^2*d^4*n - 3*(b^3*n^2 - 4*a*b^2*
n)*x^2*e^4)*log(c) + 4*((7*b^3*d*n^3 - 12*a*b^2*d*n^2)*x*e^3 + 3*(25*b^3*d^3*n^3 - 12*a*b^2*d^3*n^2)*e - 12*(3
*b^3*d^3*n^2*e + b^3*d*n^2*x*e^3)*log(c))*sqrt(x))*log(sqrt(x)*e + d) + 36*(3*(b^3*n^2 - 4*a*b^2*n + 8*a^2*b)*
x^2*e^4 + 2*(13*b^3*d^2*n^2 - 12*a*b^2*d^2*n)*x*e^2)*log(c) + 4*((37*b^3*d*n^3 - 84*a*b^2*d*n^2 + 72*a^2*b*d*n
)*x*e^3 + 72*(3*b^3*d^3*n*e + b^3*d*n*x*e^3)*log(c)^2 + 3*(415*b^3*d^3*n^3 - 300*a*b^2*d^3*n^2 + 72*a^2*b*d^3*
n)*e - 12*((7*b^3*d*n^2 - 12*a*b^2*d*n)*x*e^3 + 3*(25*b^3*d^3*n^2 - 12*a*b^2*d^3*n)*e)*log(c))*sqrt(x))*e^(-4)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x \left (a + b \log {\left (c \left (d + e \sqrt {x}\right )^{n} \right )}\right )^{3}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*ln(c*(d+e*x**(1/2))**n))**3,x)

[Out]

Integral(x*(a + b*log(c*(d + e*sqrt(x))**n))**3, x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1483 vs. \(2 (529) = 1058\).
time = 4.57, size = 1483, normalized size = 2.49 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*log(c*(d+e*x^(1/2))^n))^3,x, algorithm="giac")

[Out]

1/576*(288*b^3*x^2*e*log(c)^3 + 864*a*b^2*x^2*e*log(c)^2 + (288*(sqrt(x)*e + d)^4*e^(-3)*log(sqrt(x)*e + d)^3
- 1152*(sqrt(x)*e + d)^3*d*e^(-3)*log(sqrt(x)*e + d)^3 + 1728*(sqrt(x)*e + d)^2*d^2*e^(-3)*log(sqrt(x)*e + d)^
3 - 1152*(sqrt(x)*e + d)*d^3*e^(-3)*log(sqrt(x)*e + d)^3 - 216*(sqrt(x)*e + d)^4*e^(-3)*log(sqrt(x)*e + d)^2 +
1152*(sqrt(x)*e + d)^3*d*e^(-3)*log(sqrt(x)*e + d)^2 - 2592*(sqrt(x)*e + d)^2*d^2*e^(-3)*log(sqrt(x)*e + d)^2
+ 3456*(sqrt(x)*e + d)*d^3*e^(-3)*log(sqrt(x)*e + d)^2 + 108*(sqrt(x)*e + d)^4*e^(-3)*log(sqrt(x)*e + d) - 76
8*(sqrt(x)*e + d)^3*d*e^(-3)*log(sqrt(x)*e + d) + 2592*(sqrt(x)*e + d)^2*d^2*e^(-3)*log(sqrt(x)*e + d) - 6912*
(sqrt(x)*e + d)*d^3*e^(-3)*log(sqrt(x)*e + d) - 27*(sqrt(x)*e + d)^4*e^(-3) + 256*(sqrt(x)*e + d)^3*d*e^(-3) -
1296*(sqrt(x)*e + d)^2*d^2*e^(-3) + 6912*(sqrt(x)*e + d)*d^3*e^(-3))*b^3*n^3 + 12*(72*(sqrt(x)*e + d)^4*e^(-3
)*log(sqrt(x)*e + d)^2 - 288*(sqrt(x)*e + d)^3*d*e^(-3)*log(sqrt(x)*e + d)^2 + 432*(sqrt(x)*e + d)^2*d^2*e^(-3
)*log(sqrt(x)*e + d)^2 - 288*(sqrt(x)*e + d)*d^3*e^(-3)*log(sqrt(x)*e + d)^2 - 36*(sqrt(x)*e + d)^4*e^(-3)*log
(sqrt(x)*e + d) + 192*(sqrt(x)*e + d)^3*d*e^(-3)*log(sqrt(x)*e + d) - 432*(sqrt(x)*e + d)^2*d^2*e^(-3)*log(sqr
t(x)*e + d) + 576*(sqrt(x)*e + d)*d^3*e^(-3)*log(sqrt(x)*e + d) + 9*(sqrt(x)*e + d)^4*e^(-3) - 64*(sqrt(x)*e +
d)^3*d*e^(-3) + 216*(sqrt(x)*e + d)^2*d^2*e^(-3) - 576*(sqrt(x)*e + d)*d^3*e^(-3))*b^3*n^2*log(c) + 864*a^2*b
*x^2*e*log(c) + 72*(12*(sqrt(x)*e + d)^4*e^(-3)*log(sqrt(x)*e + d) - 48*(sqrt(x)*e + d)^3*d*e^(-3)*log(sqrt(x)
*e + d) + 72*(sqrt(x)*e + d)^2*d^2*e^(-3)*log(sqrt(x)*e + d) - 48*(sqrt(x)*e + d)*d^3*e^(-3)*log(sqrt(x)*e + d
) - 3*(sqrt(x)*e + d)^4*e^(-3) + 16*(sqrt(x)*e + d)^3*d*e^(-3) - 36*(sqrt(x)*e + d)^2*d^2*e^(-3) + 48*(sqrt(x)
*e + d)*d^3*e^(-3))*b^3*n*log(c)^2 + 12*(72*(sqrt(x)*e + d)^4*e^(-3)*log(sqrt(x)*e + d)^2 - 288*(sqrt(x)*e + d
)^3*d*e^(-3)*log(sqrt(x)*e + d)^2 + 432*(sqrt(x)*e + d)^2*d^2*e^(-3)*log(sqrt(x)*e + d)^2 - 288*(sqrt(x)*e + d
)*d^3*e^(-3)*log(sqrt(x)*e + d)^2 - 36*(sqrt(x)*e + d)^4*e^(-3)*log(sqrt(x)*e + d) + 192*(sqrt(x)*e + d)^3*d*e
^(-3)*log(sqrt(x)*e + d) - 432*(sqrt(x)*e + d)^2*d^2*e^(-3)*log(sqrt(x)*e + d) + 576*(sqrt(x)*e + d)*d^3*e^(-3
)*log(sqrt(x)*e + d) + 9*(sqrt(x)*e + d)^4*e^(-3) - 64*(sqrt(x)*e + d)^3*d*e^(-3) + 216*(sqrt(x)*e + d)^2*d^2*
e^(-3) - 576*(sqrt(x)*e + d)*d^3*e^(-3))*a*b^2*n^2 + 288*a^3*x^2*e + 144*(12*(sqrt(x)*e + d)^4*e^(-3)*log(sqrt
(x)*e + d) - 48*(sqrt(x)*e + d)^3*d*e^(-3)*log(sqrt(x)*e + d) + 72*(sqrt(x)*e + d)^2*d^2*e^(-3)*log(sqrt(x)*e
+ d) - 48*(sqrt(x)*e + d)*d^3*e^(-3)*log(sqrt(x)*e + d) - 3*(sqrt(x)*e + d)^4*e^(-3) + 16*(sqrt(x)*e + d)^3*d*
e^(-3) - 36*(sqrt(x)*e + d)^2*d^2*e^(-3) + 48*(sqrt(x)*e + d)*d^3*e^(-3))*a*b^2*n*log(c) + 72*(12*(sqrt(x)*e +
d)^4*e^(-3)*log(sqrt(x)*e + d) - 48*(sqrt(x)*e + d)^3*d*e^(-3)*log(sqrt(x)*e + d) + 72*(sqrt(x)*e + d)^2*d^2*
e^(-3)*log(sqrt(x)*e + d) - 48*(sqrt(x)*e + d)*d^3*e^(-3)*log(sqrt(x)*e + d) - 3*(sqrt(x)*e + d)^4*e^(-3) + 16
*(sqrt(x)*e + d)^3*d*e^(-3) - 36*(sqrt(x)*e + d)^2*d^2*e^(-3) + 48*(sqrt(x)*e + d)*d^3*e^(-3))*a^2*b*n)*e^(-1)

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Mupad [B]
time = 0.77, size = 840, normalized size = 1.41 \begin {gather*} {\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}^3\,\left (\frac {b^3\,x^2}{2}-\frac {b^3\,d^4}{2\,e^4}\right )-x^{3/2}\,\left (\frac {d\,\left (2\,a^3-\frac {3\,a^2\,b\,n}{2}+\frac {3\,a\,b^2\,n^2}{4}-\frac {3\,b^3\,n^3}{16}\right )}{3\,e}-\frac {d\,\left (24\,a^3-12\,a\,b^2\,n^2+7\,b^3\,n^3\right )}{36\,e}\right )-{\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}^2\,\left (\frac {x^{3/2}\,\left (\frac {b^2\,d\,\left (4\,a-b\,n\right )}{e}-\frac {4\,a\,b^2\,d}{e}\right )}{2}-\frac {3\,b^2\,x^2\,\left (4\,a-b\,n\right )}{8}+\frac {d\,\left (12\,a\,b^2\,d^3-25\,b^3\,d^3\,n\right )}{8\,e^4}+\frac {d^2\,\sqrt {x}\,\left (\frac {6\,b^2\,d\,\left (4\,a-b\,n\right )}{e}-\frac {24\,a\,b^2\,d}{e}\right )}{4\,e^2}-\frac {d\,x\,\left (\frac {6\,b^2\,d\,\left (4\,a-b\,n\right )}{e}-\frac {24\,a\,b^2\,d}{e}\right )}{8\,e}\right )+x\,\left (\frac {d\,\left (\frac {d\,\left (2\,a^3-\frac {3\,a^2\,b\,n}{2}+\frac {3\,a\,b^2\,n^2}{4}-\frac {3\,b^3\,n^3}{16}\right )}{e}-\frac {d\,\left (24\,a^3-12\,a\,b^2\,n^2+7\,b^3\,n^3\right )}{12\,e}\right )}{2\,e}+\frac {b^2\,d^2\,n^2\,\left (12\,a-13\,b\,n\right )}{16\,e^2}\right )-\sqrt {x}\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (2\,a^3-\frac {3\,a^2\,b\,n}{2}+\frac {3\,a\,b^2\,n^2}{4}-\frac {3\,b^3\,n^3}{16}\right )}{e}-\frac {d\,\left (24\,a^3-12\,a\,b^2\,n^2+7\,b^3\,n^3\right )}{12\,e}\right )}{e}+\frac {b^2\,d^2\,n^2\,\left (12\,a-13\,b\,n\right )}{8\,e^2}\right )}{e}+\frac {b^2\,d^3\,n^2\,\left (12\,a-25\,b\,n\right )}{4\,e^3}\right )+x^2\,\left (\frac {a^3}{2}-\frac {3\,a^2\,b\,n}{8}+\frac {3\,a\,b^2\,n^2}{16}-\frac {3\,b^3\,n^3}{64}\right )+\frac {\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )\,\left (\frac {x^{3/2}\,\left (16\,b\,d\,e^3\,\left (6\,a^2-b^2\,n^2\right )-12\,b\,d\,e^3\,\left (8\,a^2-4\,a\,b\,n+b^2\,n^2\right )\right )}{12\,e^2}-\frac {x\,\left (\frac {d\,\left (16\,b\,d\,e^3\,\left (6\,a^2-b^2\,n^2\right )-12\,b\,d\,e^3\,\left (8\,a^2-4\,a\,b\,n+b^2\,n^2\right )\right )}{e}-24\,b^3\,d^2\,e^2\,n^2\right )}{8\,e^2}+\frac {\sqrt {x}\,\left (\frac {d\,\left (\frac {d\,\left (16\,b\,d\,e^3\,\left (6\,a^2-b^2\,n^2\right )-12\,b\,d\,e^3\,\left (8\,a^2-4\,a\,b\,n+b^2\,n^2\right )\right )}{e}-24\,b^3\,d^2\,e^2\,n^2\right )}{e}-48\,b^3\,d^3\,e\,n^2\right )}{4\,e^2}+\frac {3\,b\,e^2\,x^2\,\left (8\,a^2-4\,a\,b\,n+b^2\,n^2\right )}{4}\right )}{4\,e^2}-\frac {\ln \left (d+e\,\sqrt {x}\right )\,\left (72\,a^2\,b\,d^4\,n-300\,a\,b^2\,d^4\,n^2+415\,b^3\,d^4\,n^3\right )}{48\,e^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a + b*log(c*(d + e*x^(1/2))^n))^3,x)

[Out]

log(c*(d + e*x^(1/2))^n)^3*((b^3*x^2)/2 - (b^3*d^4)/(2*e^4)) - x^(3/2)*((d*(2*a^3 - (3*b^3*n^3)/16 + (3*a*b^2*
n^2)/4 - (3*a^2*b*n)/2))/(3*e) - (d*(24*a^3 + 7*b^3*n^3 - 12*a*b^2*n^2))/(36*e)) - log(c*(d + e*x^(1/2))^n)^2*
((x^(3/2)*((b^2*d*(4*a - b*n))/e - (4*a*b^2*d)/e))/2 - (3*b^2*x^2*(4*a - b*n))/8 + (d*(12*a*b^2*d^3 - 25*b^3*d
^3*n))/(8*e^4) + (d^2*x^(1/2)*((6*b^2*d*(4*a - b*n))/e - (24*a*b^2*d)/e))/(4*e^2) - (d*x*((6*b^2*d*(4*a - b*n)
)/e - (24*a*b^2*d)/e))/(8*e)) + x*((d*((d*(2*a^3 - (3*b^3*n^3)/16 + (3*a*b^2*n^2)/4 - (3*a^2*b*n)/2))/e - (d*(
24*a^3 + 7*b^3*n^3 - 12*a*b^2*n^2))/(12*e)))/(2*e) + (b^2*d^2*n^2*(12*a - 13*b*n))/(16*e^2)) - x^(1/2)*((d*((d
*((d*(2*a^3 - (3*b^3*n^3)/16 + (3*a*b^2*n^2)/4 - (3*a^2*b*n)/2))/e - (d*(24*a^3 + 7*b^3*n^3 - 12*a*b^2*n^2))/(
12*e)))/e + (b^2*d^2*n^2*(12*a - 13*b*n))/(8*e^2)))/e + (b^2*d^3*n^2*(12*a - 25*b*n))/(4*e^3)) + x^2*(a^3/2 -
(3*b^3*n^3)/64 + (3*a*b^2*n^2)/16 - (3*a^2*b*n)/8) + (log(c*(d + e*x^(1/2))^n)*((x^(3/2)*(16*b*d*e^3*(6*a^2 -
b^2*n^2) - 12*b*d*e^3*(8*a^2 + b^2*n^2 - 4*a*b*n)))/(12*e^2) - (x*((d*(16*b*d*e^3*(6*a^2 - b^2*n^2) - 12*b*d*e
^3*(8*a^2 + b^2*n^2 - 4*a*b*n)))/e - 24*b^3*d^2*e^2*n^2))/(8*e^2) + (x^(1/2)*((d*((d*(16*b*d*e^3*(6*a^2 - b^2*
n^2) - 12*b*d*e^3*(8*a^2 + b^2*n^2 - 4*a*b*n)))/e - 24*b^3*d^2*e^2*n^2))/e - 48*b^3*d^3*e*n^2))/(4*e^2) + (3*b
*e^2*x^2*(8*a^2 + b^2*n^2 - 4*a*b*n))/4))/(4*e^2) - (log(d + e*x^(1/2))*(415*b^3*d^4*n^3 - 300*a*b^2*d^4*n^2 +
72*a^2*b*d^4*n))/(48*e^4)

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